scholarly journals Invariant Kähler structures on the cotangent bundles of compact symmetric spaces

2003 ◽  
Vol 169 ◽  
pp. 191-217 ◽  
Author(s):  
Ihor V. Mykytyuk
Keyword(s):  
Geodesic Flow ◽  
Symmetric Spaces ◽  
Compact Type ◽  
Reduction Procedure ◽  
Cotangent Bundles ◽  
Rank One ◽  
Compact Symmetric Spaces ◽  
Kähler Structures ◽  
Tangent Bundles

AbstractFor rank-one symmetric spaces M of the compact type all Kähler structures Fλ, defined on their punctured tangent bundles T0 M and invariant with respect to the normalized geodesic flow on T0 M, are constructed. It is shown that this class {Fλ} of Kähler structures is stable under the reduction procedure.

Ergodic theory and rigidity on the symmetric space of non-compact type

2001 ◽  
Vol 21 (1) ◽  
pp. 93-114 ◽  
Author(s):  
INKANG KIM
Keyword(s):  
Ergodic Theory ◽  
Symmetric Spaces ◽  
Isometry Group ◽  
Length Spectrum ◽  
Compact Type ◽  
Negatively Curved ◽  
Rank One ◽  
Convex Cocompact ◽  
Marked Length Spectrum ◽  
Locally Symmetric Manifolds

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

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